Babai [1]} obtained the well-known criterion for a Cayley digraph to be a CI-digraph: a Cayley digraph \(\hbox{\rm Cay}(G,S)\)  is a CI-digraph if and only if all regular subgroups of \(\hbox{\rm Aut}(\hbox{\rm Cay}(G,S))\)  isomorphic to \(G\)  are conjugate. This was also proved by Alspach and Parsons [2]}. Based on this criterion, the follow proposition is straightforward (also see [3]}).
